[Paper Review] Heavy Wilson Quarks and O($a$) Improvement: Nonperturbative Results for $b_{ m g}$

Mattia Dalla Brida, Roman Höllwieser+5 more|arXiv (Cornell University)|Dec 30, 2023

Quantum Chromodynamics and Particle Interactions60 references1 citations

TL;DR

This paper presents a nonperturbative determination of the coefficient $ b_g(g^2_0) $, which controls the mass-dependent rescaling of the bare coupling in O($a$) improved lattice QCD with heavy Wilson quarks. Using chiral Ward identities and gradient flow observables in finite volumes, the authors derive improvement conditions and compute $ b_g $ across a range of $ \beta $ values and lattice spacings, achieving a precision that reduces systematic uncertainties in decoupling studies of $ \alpha_s(m_Z) $.

ABSTRACT

With Wilson quarks, on-shell O($a$) improvement of the lattice QCD action is achieved by including the Sheikholeslami-Wohlert term and two further operators of mass dimension 5, which amount to a mass-dependent rescaling of the bare parameters. We here focus on the rescaled bare coupling, $ ilde{g}_0^2 = g_0^2(1 + b_{ m g} am_{ m q})$, and the determination of $b_{ m g}(g_0^2)$, which is currently only known to 1-loop order of perturbation theory. We derive suitable improvement conditions in the chiral limit and in a finite space-time volume and evaluate these for different gluonic observables, both with and without the gradient flow. The choice of $β$-values and the line of constant physics are motivated by the ALPHA collaboration's decoupling strategy to determine $α_s(m_Z)$. However, the improvement conditions and some insight into systematic effects may prove useful in other contexts, too.

Motivation & Objective

• To nonperturbatively determine the coefficient $ b_g(g^2_0) $, which governs the mass-dependent rescaling of the bare coupling in O($a$) improved lattice QCD with heavy quarks.
• To address the large systematic uncertainty in $ \alpha_s(m_Z) $ determinations from the ALPHA collaboration, which currently relies on a 100% assumed uncertainty in the perturbative $ b_g $ estimate.
• To establish reliable improvement conditions based on chiral symmetry restoration in finite volumes and gradient flow observables for use in heavy quark and decoupling studies.
• To validate the method through consistency checks across different observables, lines of constant physics, and lattice parameters.

Proposed method

• Derives improvement conditions using chiral Ward identities and the restoration of chiral symmetry in finite space-time volumes to constrain $ b_g $.
• Employs the gradient flow energy density and Creutz ratios as gluonic observables to extract $ b_g $, both with and without the gradient flow.
• Implements a line of constant physics (LCP) motivated by the ALPHA collaboration's decoupling project, with $ \Lambda_{\overline{MS}} $ fixed and $ \beta $ values spanning $ 4.12 \leq \beta \leq 19.46 $.
• Performs simulations with $ N_f = 3 $ degenerate quarks using the tree-level O($a^2$) improved Lüscher-Weisz gauge action and nonperturbative O($a$) improvement.
• Uses rational approximations for the $ \beta $-dependence of $ \sigma(0.18) $ and $ \hat{\chi} $, and computes derivatives via $ \partial \sigma / \partial g^2_0 $ and $ \partial \hat{\chi} / \partial g^2_0 $ from simulations at different $ \beta $ and $ \kappa $ values.
• Applies a relaxation of the LCP condition for $ \beta \geq 4.9 $ to improve statistical precision and stability in the $ b_g $ determination.

Experimental results

Research questions

• RQ1 Can $ b_g(g^2_0) $ be nonperturbatively determined using chiral Ward identities and finite-volume observables in lattice QCD with heavy Wilson quarks?
• RQ2 How do the results for $ b_g $ from gradient flow and Creutz ratio observables compare, and what consistency checks can be performed?
• RQ3 To what extent do the improvement conditions derived in the chiral limit and finite volume yield stable and reliable $ b_g $ values across varying lattice spacings and $ \beta $ values?
• RQ4 How do the nonperturbative results for $ b_g $ compare with the 1-loop perturbative estimate $ b_g = 0.012 \times N_f g^2_0 $, especially in the large-$ \beta $ (small-$ g^2_0 $) regime?
• RQ5 Can the method be applied to reduce systematic errors in $ \alpha_s(m_Z) $ determinations that currently rely on a 100% uncertainty in the perturbative $ b_g $ estimate?

Key findings

• The nonperturbative determination of $ b_g(g^2_0) $ is achieved across a wide range of $ \beta $ values from 4.12 to 19.46, with lattice spacings covering $ a \sim 0.13 $ to $ 0.013 $ fm.
• The results for $ b_g $ from gradient flow energy density and Creutz ratios show good consistency across different $ \beta $ values and volumes, validating the improvement conditions.
• For $ \beta \geq 4.9 $, relaxing the line of constant physics condition improves statistical precision and stability in the $ b_g $ determination, particularly in the small-$ g^2_0 $ regime.
• The nonperturbative $ b_g $ values are found to be significantly larger than the 1-loop perturbative estimate $ b_g = 0.012 \times N_f g^2_0 $, especially at larger $ \beta $, indicating a nontrivial nonperturbative contribution.
• The method successfully reduces systematic uncertainties in $ b_g $, which directly impacts the precision of $ \alpha_s(m_Z) $ determinations in the ALPHA decoupling project.
• The derived improvement conditions and observables are robust across different simulation parameters and provide a viable framework for future nonperturbative $ b $-coefficient determinations in heavy quark and decoupling studies.